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Derivation

1. Initial program 18.1

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]

2. Initial simplification1.4

   \[\leadsto \frac{\frac{t1}{t1 + u}}{\frac{t1 + u}{-v}}\]
3. Using strategy rm

4. Applied div-inv1.4

   \[\leadsto \frac{\frac{t1}{t1 + u}}{\color{blue}{\left(t1 + u\right) \cdot \frac{1}{-v}}}\]

5. Applied *-un-lft-identity1.4

   \[\leadsto \frac{\color{blue}{1 \cdot \frac{t1}{t1 + u}}}{\left(t1 + u\right) \cdot \frac{1}{-v}}\]

6. Applied times-frac1.4

   \[\leadsto \color{blue}{\frac{1}{t1 + u} \cdot \frac{\frac{t1}{t1 + u}}{\frac{1}{-v}}}\]

7. Simplified1.4

   \[\leadsto \frac{1}{t1 + u} \cdot \color{blue}{\left(\frac{-v}{u + t1} \cdot t1\right)}\]
8. Using strategy rm

9. Applied associate-*l/1.3

   \[\leadsto \color{blue}{\frac{1 \cdot \left(\frac{-v}{u + t1} \cdot t1\right)}{t1 + u}}\]

10. Simplified1.5

    \[\leadsto \frac{\color{blue}{\frac{t1}{\frac{u + t1}{-v}}}}{t1 + u}\]

11. Final simplification1.5

    \[\leadsto \frac{\frac{t1}{\frac{u + t1}{-v}}}{u + t1}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018285 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))